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Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"18baaa3c7d8486647cda7fb0ec111c2d154104b95a4029af77a59ceace91bdd7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ef0a5df5e949a196","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1,      9;\n  3: 1,     51;\n  4: 1,    225,      333;\n  5: 1,    927,     5184;\n  6: 1,   3741,    53649,     32103;\n  7: 1,  15003,   468108,    980370;\n  8: 1,  60057,  3738069,  18390591,   5978529;\n  9: 1, 240279, 28373544, 275023890, 301349079;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(6*n-6)*x)*P(n-1,x) + (3*x-12*x^2)*d/dx P(n-1,x) + 3*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (6*n - 6)*x)*P[n - 1, x] + (3*x - 12*x^2)*D[P[n - 1, x], x] + 3*x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + (6*n - 6)*x)*v[i - 1] + (3*x - 12*x^2)*deriv(v[i - 1], x) + 3*x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ef0a5df5e949a196","steps":["\u03b3"],"edges":["e-53cca568857b5184"],"text":"A144440 \u2192 \u03b3"},"headline":"A144440 \u2192 \u03b3","chain":null}